Impossible figures are fascinating objects, related to
art, psychology, and mathematics [1]. Lionel and Roger
Penrose (father and son) introduced the impossible tribar in 1958
[2] (Fig. 1). A figure is called impossible
when "a contradiction in our interpretation is noticed but does not result
in our rejecting it in favour of a consistent one" [3].
The object represented in Fig. 1 is an impossible figure
because our mind tries to interpret it as a three-dimensional (3D) object
in the Euclidean space, with straight edges and planar faces, instead of
interpreting it, for example, as a two-dimensional object drawn on the
paper plane (which is perfectly possible).

Fig. 1

Impossible figures have inspired researchers with more
than one hundred papers (see Kulpa [4] for an extensive
bibliography), and the Dutch artist Escher [5] with some
famous drawings (see e.g. Fig. 2).

We define a multibar as a polyhedron [7],[8]
that is any four-fold cover of a regular polygon. The first example is
Penrose's tribar (Fig. 1), but other three-bars can be
constructed as four-fold covers of an equilateral triangle. Some examples
are shown in Fig. 3.

Fig. 3

Intuitively, the two figures on the left seem impossible
while the third one seems possible. We will now consider the family derived
from the tribar of Fig. 1 (see Fig. 4).

It appears immediately that there are two kinds of edges:
the longitudinal edges (drawn in blue) and the transversal
edges (drawn in green). We were inspired by the work of T. M. Cowan [9],[10]
who used braid theory to analyse impossible figures. He described each
face of the impossible polyhedron by the string of a braid. We find more
natural to describe each longitudinal edge of the impossible polyhedron
by the string of a braid. Our aim is to show that braid theory is not sufficient
to describe a multibar. The braid has to be closed into a knot or link
in order for the multibar to have ony one representative.

3. Braids

Let us introduce some basics of braid theory. A n-string
braid consists of n strings going conventially from top to bottom.
One string must never turn upside. The strings may cross each other and,
at each crossing, the string passing behind the other is indicated by a
gap. We use P. Dehornoy's convention [11] using lower-case
and upper-case letters instead of si and si-1
to designate the crossings (see Fig. 5).

Fig. 6

The braids form a group under concatenation, with the
following relations (see Fig. 7).

Fig. 7

The first relation (ac = ca) is valid for
any two letters distant of at least 2 in the alphabet. The second relation
(aba = bab) is valid for any two consecutive letters in the
alphabet. One also have to take into account the usual inverse property
of any group, i.e., aA = bB = cC = identity. The identity
braid with n strings corresponds to n descending strings
that don't cross.

Let us now look at one corner of a polyhedron of Fig.
5 (they all are identical). The corresponding 4-string braid is drawn
in Fig. 8.

Fig. 8

The 4-string braids corresponding to the tribar, the four-bar
and the five-bar displayed in Fig. 5 are thus, respectively,
bacbacbac,
bacbacbacbac,
and bacbacbacbacbac.

Now, we would like to draw your attention to the following
fact. The polyhedra of Fig. 4 could as well have been drawn as in Fig.
9.

Fig. 9

A corner of such a polyhedron would be represented by
the following 4-string braid (Fig. 10)

Fig. 10

The 4-string braids corresponding to the tribar, the four-bar
and the five-bar of Fig. 4 would have been, respectively,
acbacbacb,
acbacbacbacb,
and acbacbacbacbacb.

4. Knots and links

We now come to the closure of a braid, represented in
Fig.
11. It gives rise to a knot (resp., link), that is, the embedding of
one (resp., several) closed curve(s) in 3D space.

Fig. 11

A theorem of A. A. Markov [12] states
that different braids correspond to the same knot/link after closure if
they are related by a finite series of two moves:

(i) conjugation (as is the case for the multibars we consider
here: the b crossing is "moved" from one end of the braid to the
other end along the closure),

(ii) move where a string is added together with a crossing
between the last two strings. This move is off context here since the number
of strings is fixed to 4.

We thus see that the multibars drawn in Fig.
4 have only one knot/link representative each, independently of the
angle of view taken by the observer.

The superiority to the knot/link vision versus the braid
vision is even more manifest when we consider the "magic pentagram" (Fig.
12, left), an impossible figure due to L. Sallows [13]
and displayed on the last page of F. J. Swetz's book [14].
It is a four-fold cover of a pentagram and thus obeys our definition of
multibar.

Fig. 12

A 4-string braid representative of the magic pentagram
would be bacbacbacbacbac or acbacbacbacbacb, thus undistinguishable
from the five-bar of Fig. 4. On the contrary, the link
corresponding to the magic pentagram is different from the link correponding
to the regular five-bar since the embedding in the 3D space is taken into
account. It is a satellite link [15] whose companion
knot is the torus knot with 5 crossings (Fig. 12, right).
Such a satellite link may not be trivial since its companion is a non-trivial
knot.

On the countrary, if we draw (Fig. 13)
the hidden edges of the three-bar of Fig. 3, right, which
seems not to be an impossible figure, we see that the representative link
is a trivial link with four components.

Fig. 13

We finally come to the following
tentative criterion in order for a multibar to be an impossible figure:
"If the knot or link corresponding to a multibar is different from the
trivial link with four components, then the multibar is an impossible figure."

Acknowledgment

All my gratitude goes to Francis Buekenhout who made me
discover the "magic pentagram" and who incited me to write this text.